Before Battles
First thing you should do is know your Yo-kai. Go to their pages and look at their stats. Look at what is lacking and what is stronger, as well as their attitude and how much they are prone to loaf. It's better to know it as soon as you get them so you can use the according book to change their attitudes and raise specifics stats. There are already info on it around this sub!
Then go to their moves page and get to know it too. Most people don't read it and they don't really know how much damage their Yo-kai can do and what are the elements their moves are. Some Yo-kai can have two different moves with different elements, that's something you should check when changing attitudes as well.
Use those books that level-up specific moves. The higher the level, the bigger the damage.
Know the skills of Your yo-kai and make use of that. Some yo-kai will affect their friends and some their foes. It's better to know what are your skills when placing them on the wheel as well so you can choose which you want in battle by turning it
Make better use of the items. There are special items like the Vampiric Fangs in YW 1 and the Revenge Soul in YW 2 that will heal your Yo-kai a bit for every physical hit he does. Don't give it to a Yo-kai that does elemental damage...
Make use of the Unity Bonus, they are great. Bonuses are always great. You should try having more than one unity at the time so you can spin the wheel and get a new bonus.
During Battles
READ THE BATTLE NARRATIONS! They appear at the top of the top screen every time something happens. It will tell you there if a Yo-kai is hitting himself or a friend when inspirited.
Usually when you start a battle against a group of Yo-kai, you should use the target button so you don't attack randomly. Try to pin the one you know is the weakest against the Yo-kai you have in the front. Be it because they are from a weaker level or if you know your elemental attacks do critical damage against said Yo-kai.
It's a good strategy to have at least one Yo-kai from the Slippery Tribe that inspirits enemies and have it's attitude changed so he does it ALL THE TIME. Once he inspirits someone, PIN THEM so your Soul Meter is filled faster.
The more you let a Yo-kai use a move, the faster it level-up. Let them battle by themselves and get better at it. It's always better to have two strong Yo-kais with high level moves hitting an enemy while you are charging a Soultimate. You can't always rely on your Soultimates.
Once you get inspirited, try to purify them ASAP! Some bosses you keep doing it and you do not want to end up with a whole team inspirited. That's not fun at all and very difficult to get out of, if you are not very fast. And you get EXP once you purify them so it's always better to.
The tapping mini-game to purify Yo-kai has a weak spot. Try to find the place where you tap in which the sound effect and the visual cue is different. It breaks the glass more easily if you keep tapping at that place and you end it faster.
Use your talismans. Specially before a Soultimate. Use the spirit one to get a bigger elemental damage, if the Soultimate uses an element. Use a strength one if it's a physical one. Use the speed one if you want a Yo-kai to attack faster. Use the defense one on a Yo-kai that's too weak or on a healer, so you keep it around a bit more.
The boss Yo-kais always have weak spots. If they don't have multiple, they will have some sort of interaction that will either make your life easier or actually allow you to defeat them... Otherwise you would just be there for hours and hours. Try to look at everything with the target pin. If it glows red, you can pin there. It's important to always try to find the weak spots right when you start a battle or use a multi-hit Soultimate and pay attention where the damage shows up.
Keep an eye out for the wisps. I know it's a lot to look at sometimes but if you get a wisp to recharge that Soultimate meter instantly, you are one step ahead already!
Don't use your food only to befriend yo-kai. You should heal them if you think it's needed. The game tells you how much each item heals on the bottom of the top screen. Make good choices...
PvP Matches
There's a timer to rotate your wheel. Only do it if you really need. You might want to turn it when your enemy is trying to use a Soultimate attack.
Your opponent doesn't know all your Yo-kai. You can hide your stronger Yo-kais at the back and wait till your opponent is weak to rotate... Just remember they too can be hiding something from you as well. The Soultimate meter isn't charged initially on PvP so try to inspirit someone and charge your meter faster.
The key to the PVP matches is to know the Yo-kai so if you know a Yo-kai is weaker against your team, GO FOR IT. Target that damn thing and take them out one by one. If your Yo-kai are trained well, you won't have much surprises.